A Study of Clight and its Semantics

CompCert is a certified C compiler which comes with a proof of semantics preservation. What this means is the following: the semantics of the C code you write is preserved by CompCert compilation passes up to the generated machine code. I had been interested in CompCert for quite some times, and ultimately challenged myself to study Clight and its semantics. This write-up is the result of this challenge, written as I was progressing.

1. Installing CompCert
2. Problem Statement
3. Proof Walkthrough
4. Conclusion

Installing CompCert

CompCert has been added to opam , and as a consequence can be very easily used as a library for other Coq developments. A typical use case is for a project to produce Clight (the high-level AST of CompCert), and to benefit from CompCert proofs after that. Installing CompCert is as easy as

opam install compcert

Once opam terminates, the compcert namespace becomes available. In addition, several binaries are now available if you have correctly set your PATH environment variable. For instance, clightgen takes a C file as an argument, and generates a Coq file which contains the Clight generated by CompCert.

Problem Statement

Our goal for this first write-up is to prove that the C function

int add (int x, int y) {
    return x + y;
}

returns the expected result, that is x + y . The clightgen tool generates (among other things) the following AST (note: I have modified it in order to improve its readability).

The fields of the function type are pretty self-explanatory (as it is often the case in CompCert’s ASTs as far as I can tell for now). Identifiers in Clight are (positive) indices. The fn_body field is of type statement, with the particular constructor Sreturn whose argument is of type option expr, and statement and expr look like the two main types to study. The predicates step1 and step2 allow for reasoning about the execution of a function, step by step (hence the name). It appears that clightgen generates Clight terms using the function call convention encoded by step2. To reason about a complete execution, it appears that we can use star (from the Smallstep module) which is basically a trace of step. These semantics are defined as predicates (that is, they live in Prop). They allow for reasoning about state-transformation, where a state is either

• A function call, with a given list of arguments and a continuation
• A function return, with a result and a continuation
• A statement execution within a function

We import several CompCert modules to manipulate values (in our case, bounded integers).

Putting everything together, the lemma we want to prove about f_add is the following.

Proof Walkthrough

We introduce a custom inversion tactic which does some clean-up in addition to just perform the inversion.

We can now try to prove our lemma.

We first destruct trace, and we rename the generated hypothesis in order to improve the readability of these notes.

This generates two hypotheses.

Hstep : step1
          env
          (Callstate (Ctypes.Internal f_add)
                     [Vint x; Vint y]
                     Kstop
                     m)
          t1
          s2
Hstar : Smallstep.star
          step2
          env
          s2
          t2
          (Returnstate (Vint z) Kstop m')

In other words, to “go” from a Callstate of f_add to a Returnstate, there is a first step from a Callstate to a state s2, then a succession of steps to go from s2 to a Returnstate. We consider the single step, in order to determine the actual value of s2 (among other things). To do that, we use smart_inv on Hstep, and again perform some renaming.

This produces two effects. First, a new hypothesis is added to the context.

f_entry : function_entry1
            env
            f_add
            [Vint x; Vint y]
            m
            c_env
            tmp_env
            m1

Then, the Hstar hypothesis has been updated, because we now have a more precise value of s2. More precisely, s2 has become

State
  f_add
  (Sreturn
    (Some (Ebinop Oadd
                  (Etempvar _x tint)
                  (Etempvar _y tint)
                  tint)))
  Kstop
  c_env
  tmp_env
  m1

Using the same approach as before, we can uncover the next step.

The resulting hypotheses are

Hstep : step2 env
          (State
             f_add
             (Sreturn
               (Some
                 (Ebinop Oadd
                 (Etempvar _x tint)
                 (Etempvar _y tint)
                 tint)))
             Kstop c_env tmp_env m1) t1 s2
Hstar : Smallstep.star
          step2
          env
          s2
          t0
          (Returnstate (Vint z) Kstop m')

An inversion of Hstep can be used to learn more about its resulting state… So let’s do just that.

The generated hypotheses have become

res : val
ev : eval_expr env c_env tmp_env m1
       (Ebinop Oadd
               (Etempvar _x tint)
               (Etempvar _y tint)
               tint)
       res
res_equ : sem_cast res tint tint m1 = Some v'
mem_equ : Memory.Mem.free_list m1
                               (blocks_of_env env c_env)
            = Some m'0

Our understanding of these hypotheses is the following

• The expression _x + _y is evaluated using the c_env environment (and we know thanks to binding the value of _x and _y), and its result is stored in res
• res is cast into a tint value, and acts as the result of f_add

The Hstar hypothesis is now interesting

Hstar : Smallstep.star
          step2 env
          (Returnstate v' Kstop m'0) t0
          (Returnstate (Vint z) Kstop m')

It is clear that we are at the end of the execution of f_add (even if Coq generates two subgoals, the second one is not relevant and easy to discard).

We are making good progress here, and we can focus our attention on the ev hypothesis, which concerns the evaluation of the _x + _y expression. We can simplify it a bit further.

In a short-term, the hypotheses fetch_x and fetch_y are the most important.

fetch_x : eval_expr env c_env tmp_env m1 (Etempvar _x tint) v1
fetch_y : eval_expr env c_env tmp_env m1 (Etempvar _y tint) v2

The current challenge we face is to prove that we know their value. At this point, we can have a look at f_entry. This is starting to look familiar: smart_inv, then renaming, etc.

We are almost done. Let’s simplify as much as possible fetch_x and fetch_y. Each time, the smart_inv tactic generates two suboals, but only the first one is relevant. The second one is not, and can be discarded.

We now know the values of the operands of add. The two relevant hypotheses that we need to consider next are add_op and res_equ. They are easy to read.

add_op : sem_binarith
           (fun (_ : signedness) (n1 n2 : Integers.int)
              => Some (Vint (add n1 n2)))
           (fun (_ : signedness) (n1 n2 : int64)
              => Some (Vlong (Int64.add n1 n2)))
           (fun n1 n2 : Floats.float
              => Some (Vfloat (Floats.Float.add n1 n2)))
           (fun n1 n2 : Floats.float32
              => Some (Vsingle (Floats.Float32.add n1 n2)))
           v1 tint v2 tint m1 = Some res

• add_op is the addition of Vint x and Vint y, and its result is res.

res_equ : sem_cast res tint tint m1 = Some (Vint z)

• res_equ is the equation which says that the result of f_add is res, after it has been cast as a tint value

Surprisingly, neither add_op nor res_equ can be reduced easily… After some digging, I have found that this is because both rely on Archi.ptr64, which is basically opaque as far as reduction strategies are concerned. Now, there are probably more elegant (idiomatic) way to deal with this issue, but a working approach is to unfold until Archi.ptr64 appears, then use the Archi.splitlong_ptr32 lemma to replace it with its value (false).

We can now simplify add_op and res_equ, and this allows us to conclude.

Conclusion

The main difficulty of this exercise was the opaqueness of Archi.ptr64; this is surprising, since it means my struggle was technical, not conceptual. On the contrary, the definitions of Clight are easy to understand, and the CompCert documentation is very pleasant to read. Understanding Clight and its semantics can be very interesting if you are working on a language that you want to translate into machine code. However, proving functional properties of a given C snippet using only CompCert can quickly become cumbersome. From this perspective, the VST project is very interesting, as its main purpose is to provide tools to reason about Clight programs more easily.